Application of Magnetorheological Damper in Aircraft Landing Gear: A Systematic Review

Abstract

During takeoff and landing, aircraft operate in a variety of situations, posing significant challenges to landing gear systems. Passive hydraulic–pneumatic dampers are commonly used in conventional landing gear to absorb impact energy and reduce vibration. However, due to their fixed damping characteristics and inability to adjust to changing operating conditions, these passive systems have several limitations. Recent research has focused on creating intelligent landing gear systems with magnetic dampers (MR) to overcome these limitations. By changing the magnetic field acting on the MR fluid, MR dampers provide semi-active control of the landing gear dynamics and adjust the damping force in real time. This flexibility reduces structural load during landing, increases riding comfort, and improves energy absorption efficiency. This study examines the current state of MR damper application for aircraft landing gear. The review categorizes current control techniques and highlights the structural integration of MR dampers in landing gear assemblies. Purpose: The magnetorheological (MR) damper has become a promising semiactive system to replace the conventional passive damper in aircraft landing gear. However, the mechanical structure and control strategy of the MR damper must be designed to be suitable for aircraft landing gear applications. Methods: Researchers have explored the potential structure designed, the mathematical model of the MR landing gear system, and the control algorithm that was developed for aircraft landing gear applications. Results: According to the mathematical model of the MR damper, three types of models, which are pseudo-static models, parametric models, and unparameterized models, are detailed with their application. Based on these mathematical models, many control algorithms were studied, from classical control, such as PID and skyhook control, to modern control, such as intelligent control and SMC control.

1. Introduction

The aircraft landing gear is a fundamental system that supports the aircraft during takeoff and landing, and on the ground-taxiing [1]. The most functional part of the landing gear is the shock absorber, which is designed to absorb the impact forces during landing and maintain the aircraft’s stability and control on the ground [2,3]. When the aircraft lands, especially during the final approach and touchdown phase, the shock absorbers are subjected to the maximum landing impact force [4,5]. Their primary function is to absorb the substantial vertical forces resulting from the aircraft’s descent and touchdown, ensuring that these forces are safely dissipated and do not damage the aircraft structure [6]. During taxiing, an aircraft moves at relatively slow speeds, so the shock absorbers are not experiencing the high forces they would during landing. However, they still serve an important role in absorbing vibrations from the runway or taxiway surface. The shock absorbers help smooth out the ride by dampening minor shocks from surface irregularities, keeping the aircraft steady, and preventing excessive wear on the landing gear components [7]. Moreover, in some types of aircraft, the nose landing gear involves the shimmy phenomenon [8,9,10]. So, the shimmy damper is equipped in the nose landing gear [11,12,13].

Many types of shock absorbers have been developed and applied, including mechanical springs [14,15], pneumatic, liquid springs [16], and oleo-pneumatic dampers [17,18]. The oleo-pneumatic type is the most common arrangement among the different kinds of shock absorbers used on aircraft. The typical configuration of a conventional oleo-pneumatic, or passive damper, is detailed in Figure 1. The damper comprises an air chamber and an oil chamber, enclosed by a piston and a cylinder. While the air chamber focuses on a chamber on top of the cylinder, the oil has two chambers that connect through the orifice. The cross-section of the orifice is modified by the metering pin, which has a conical shape [19]. It leads to an adjustment in the viscous force, so the damper can modify a small characteristic. Thus, the landing gear is hard to modify in different landing scenarios [20].

To replace the conventional damper, NASA has developed an active damper system equipped with a strong hydraulic pump [21]. As can be seen in Figure 2, the strong hydraulic pump generates direct damping forces to resist the outside load. So, this system can control a huge range of damping forces. Following this idea, many other researchers have attempted to develop the active controller system and have had some success. Zarchi and Attaran [22] designed an active landing gear for a passenger aircraft using a multi-objective optimization technique. Sivakumar and Haran [23] analyzed a full aircraft with active landing gears by numerical simulations on a random runway profile. Yazici and Sever [24,25] applied an observer-based optimal state-feedback controller with pole location constraints to an active vibration mitigation problem in an aircraft system. Pirooz and Fateh [26] developed a robust force and displacement control for active landing gear to reduce vibration at touchdown and during taxiing [27]. In all previous research on fully active landing gear, only theories were developed, and simulations were performed using commercial software to verify the proposed theories. The biggest challenge is to develop a strong pump with high pressure and accuracy due to the huge landing impact when the aircraft makes contact with the ground. The strong pump required a huge amount of energy and space because of the large size of the electrical motor. Moreover, the servo valve is also very expensive. Thus, there is no research involved to verify this system through a real test experiment.

Instead of using the active damper, many researchers have been focusing on the semi-active damper system to replace the conventional passive damper. Karnopp [28] published the first idea of semi-active control systems that utilize the motion of the structure itself to develop control forces, the magnitude of which can be adjusted by the external power source. These systems typically require a small external power source, such as a battery, to operate [29,30]. The capacity to dynamically change the characteristics of passive energy dissipation devices, such as dampers or springs, in real-time sets semi-active control systems apart. Because of their versatility, they can perform at a level that frequently competes with fully active control systems while using a lot less energy [31]. Figure 3 details the difference in the boundary of the damping force between the passive damper, active damper, and semi-active damper. The relationship between the damping force and stroke velocity is a curve, meaning that there is only one value of damping force at one position of the damper. While the active damper can adapt the damping force independently of the damper kinematic, that is, a full rectangle as in Figure 3. The semi-active can only activate the damping force during the positive part of the stroke velocity; it occupies half of this rectangle, so it is called a semi-active damper.

The most promising way to apply the semi-active damper in aircraft landing gear is to use the magnetorheological (MR) damper [32,33,34] and the electrorheological (ER) damper [35,36,37,38]. When an electric field is applied to an ER damper, the viscosity changes. So, the damping force of an ER damper can be controlled by adjusting the intensity of this electric field [39,40]. In contrast, MR dampers use magnetorheological (MR) fluids, which exhibit a change in viscosity when subjected to a magnetic field [34,41]. The damping force is controlled by altering the strength of the magnetic field, typically by adjusting the input current to an electromagnet [42]. The ER required a high voltage range from 1 kV to 3 kV; however, it did not have a high dynamic yield stress in a compact size. MR damper’s structure is often simple, and it is controlled using a small amount of energy. Moreover, the MR damper is one of the best semi-active dampers since it can provide quick and precise adjustments [43,44,45,46]. Thus, the MR damper is often chosen to be applied to aircraft landing gear [47,48].

The design and control of landing systems equipped with MR dampers have been the subject of extensive research over the past 25 years. Research has focused on topics such as nonlinear dynamic modeling, damper design optimization, and the creation of sophisticated semi-active control schemes. For practical implementation in aircraft landing gear, many issues remain regarding the design structure and control algorithm. This review aims to provide a detailed summary of recent developments in MR landing gear systems. The paper provides an overview of significant advances in MR damper design and control strategies. In addition, the paper highlights the potential of MR technology to enable next-generation adaptive landing gear systems with improved safety, comfort, and reliability, and discusses current limitations and future research goals.

A systematic search was conducted between 2000 and 2025 across key academic databases, such as Scopus, Web of Science, IEEE Xplore, and ScienceDirect, to obtain thorough coverage of the pertinent literature, as can be seen in Appendix A. These databases were chosen because of their thorough indexing of engineering, materials science, and aerospace articles. To maximize the retrieval of relevant studies, the search method combined controlled vocabulary with free-text phrases, including “aircraft landing gear” AND “magnetorheological damper,” and “aircraft semi-active landing gear”. The reference management program Zotero was then used to organize and screen all of the search results, as detailed in Figure A1. Even though the search was thorough, it is possible that pertinent research written in other languages was missed because it was limited to English-language journals. Furthermore, even though the use of particular keywords and Boolean combinations is methodical, there is a chance that papers using different terminology or uncommon descriptors for magnetorheological dampers and landing gear systems will be overlooked.

2. Magnetorheological (MR) Damper

Magnetorheological (MR) fluid, which is a smart material, was first developed by Jacob Rainbow in the 1940s [49]. A magnetorheological damper is a damper filled with magnetorheological fluid, which is controlled by a magnetic field, usually using an electromagnet [50]. This allows the damping characteristics of the shock absorber to be continuously controlled by varying the power of the electromagnet [51]. The main components of a Magnetorheological (MR) damper work in conjunction to provide controllable damping by utilizing the unique properties of MR fluid [52]. While specific designs may vary, the fundamental components include MR fluid, piston, cylinder, spring, and orifice, as shown in Figure 4.

MR fluids are colloidal suspensions, that is, they consist of small, micron-sized ferromagnetic or paramagnetic particles (usually iron) dispersed in a non-magnetic carrier fluid, such as mineral oil, silicone oil, or synthetic hydrocarbons [53]. The particles are often coated to prevent agglomeration and sedimentation. In the absence of an external magnetic field, these particles are randomly distributed throughout the carrier fluid. The fluid behaves much like a conventional hydraulic fluid, with a relatively low viscosity that allows for easy flow [54]. When an external magnetic field is applied to the MR fluid, the ferromagnetic particles become magnetized and align themselves almost instantaneously along the lines of magnetic flux [55]. This causes them to form chain-like structures that span the fluid gap. These chains are stiff and resistant to deformation. The formation of these bead chains significantly increases the resistance to fluid flow [56]. This phenomenon is perceived as a rapid and reversible increase in the apparent viscosity of the MR fluid. It essentially transforms from a free-flowing liquid to a semi-solid or gel-like state [57]. More precisely, MR fluids generate “yield stress,” which is a certain force, or shear stress, that must be overcome before the fluid begins to flow [58]. The magnitude of this yield stress is proportional to the strength of the applied magnetic field. In an MR damper, the fluid flows through channels or orifices. When a magnetic field is applied to these channels, the formation of chains of beads within the fluid creates resistance to its flow. This resistance translates directly into a damping force, which opposes the motion of the piston in the damper, as can be seen in Figure 5 [59].

The primary function of Magnetorheological (MR) dampers relies on MR fluids, whose viscous properties can be rapidly and reversibly controlled by an external magnetic field [60]. MR fluid consists of magnetic particles, a carrier liquid, and additives, all of which influence its viscosity [61,62]. Magnetic particles, typically micron-sized (approximately 0.1–10 µm) or nanometer-sized, are usually made of soft magnetic materials such as carbonyl iron (CI) [63]. These particles significantly affect viscosity under a magnetic field; factors such as particle volume fraction, size, and morphology increase flow resistance and compression stress [64,65]. Micro–nano composite systems can reduce zero-field viscosity [66]. The carrier fluid—commonly silicone, mineral, or synthetic oil—suspends the magnetic particles [67]. It largely determines dynamic viscosity and responds to temperature changes; higher temperatures reduce viscosity due to increased molecular spacing and decreased internal friction [68]. Consequently, as the temperature rises, the damping force decreases [69]. Additives, including anti-settling agents and nonferromagnetic particles such as glass powder, are incorporated to enhance stability and adjust viscosity [70]. While additives improve sedimentation stability, they often reduce the MR effect [66,71].

3. Application of MR Technology in Aircraft Landing Gear

The magnetorheological (MR) damper is typically manufactured by the LORD Corporation [59]. Commercially available magnetorheological (MR) dampers typically generate a maximum damping force of around 2.5 kN and feature a relatively short stroke length of approximately 60 mm. This output is two to three times lower than the damping force required for a typical aircraft application. Consequently, redesigning the damper is essential to meet the performance demands of landing conditions. Numerous aircraft MR landing gear designs have been developed to address this challenge, as given in

Table 1. List the design of the MR damper in the landing gear.

		Powell et al. [72,73]	Mikułowski and Holnicki-Szulc [74]	D C Batterbee et al. [75,76,77]	Saleh et al. [78,79,80]	Liu et al. [81]	Khani [82]	Han et al. [83,84]	Kang et al. [85,86,87]
Aircraft weight		582 kg	60 kg	473 kg	2260 kg	300 kg	27,397 kg	200– 250 kg	640– 720 kg
Maximum Stroke		55 mm	37 mm	120 mm	35 mm	45 mm	250 mm	200 mm	250 mm
Damping force		18.1 kN	1 kN	12–18 kN	7–11 kN	13–16 kN	60–80 kN	8–10 kN	20–25 kN
Maximum Sink speed		7.9 m/s	-	3 m/s	5 m/s	1.5 m/s	3.2 m/s	3.01 m/s	3 m/s
Manufacturing special	Gap size	0.79 mm	-	0.59 mm	0.8 mm	0.56 mm	0.85 mm	1 mm	1.3 mm
	Number of coils	3	1	1	2	1	1	2	3
	Active length	33 mm	-	28.9 mm	16 mm	-	24 mm	48 mm	49.4 mm
Application		Skids helicopter of Iron Bird	Scale Prototype Aircraft landing gear	Institute of Aviation’s I-23 aircraft	Skids helicopter	Scale Prototype aircraft landing gear	Navy A6-Intruder Aircraft landing gear	Scale Prototype of Beechcraft Baron Aircraft landing gear	Beechcraft Baron Aircraft landing gear
Electrical current		0–4 A	0–1 A	0–2.6 A	0–0.8 A	0–0.5 A	0–2 A	0–1 A	0–3 A
Goal		Maintain a constant damping force of 4000 lbf within a sink speed range of 6–26 ft/s	Reduce the impact forces by 30%	Achieve 90% shock absorber efficiency	Generate the desired damping force without violating established design constraints.	Improve the shock absorber efficiency	Improve damping efficiency by 12.3% for a sink speed of 3.2 m/s	Improve landing efficiency by 15%	Improve by 17.9% over the efficiency achieved with existing passive damping.

.

Studies range from small-scale models in the laboratory with presented masses of 60 kg to large commercial aircraft weighing 27,397 kg in simulation models, with damping forces ranging from 1 kN to 80 kN. Maximum stroke lengths vary widely, with some designs reaching 250 mm. Most systems are optimized for sink speeds of around 3 m/s, following the standard of FAR part 25 [4,88]. Working clearance sizes, when specified, range from 0.56 mm to 1 mm, which is important for MR fluid control. Current requirements vary depending on the damper size and control strategy, ranging from 0 to 0.5 A to 0–4 A. While most designs are for conventional aircraft landing gear, some are specifically designed for helicopter skids, reflecting the versatility of MR technology in diverse aviation contexts. However, there is no application of magnetorheological in UAV due to the huge weight of the MR damper [89].

The design of aircraft MR landing gear can be categorized into three main classifications, as illustrated in Figure 6. The first design is a structure similar to that developed by the Lord Company, described in Section 2. This configuration is particularly suitable for scale prototypes used to verify the dynamic modeling [74] and test the control algorithms [81]. The second design is derived from the conventional oleo-pneumatic passive damper. In this case, the metering pin is replaced by an MR valve to enable modification of the damping force. This design not only functions as a passive damper but also adapts its characteristics to various landing scenarios; however, applying current to the coil presents a significant challenge [75,76,77]. The third design represents a major advancement over the passive damper by improving the recoil valve and repositioning the air chamber. In most MR landing gear structures, the air chamber is located at the top of the cylinder, which complicates connection to the power source. By contrast, the new structure proposed by Kang et al. [85,86,87] places the air chamber within the piston, facilitating easier connection of electrical cables, though the manufacturing process becomes more complex.

Aircraft MR landing gear assemblies are highly complex systems that require extensive design and optimization efforts. Numerous tools and experimental procedures are essential for achieving optimal performance. The advent of advanced modeling tools and sophisticated computational techniques has transformed this process, enabling comprehensive simulation, multidisciplinary design optimization, and rapid prototyping. These advancements significantly enhance design efficiency, innovation, and reliability. MATLAB is widely utilized for developing mathematical models [83], optimizing designs [94], and implementing control algorithms [93]; however, its effective use demands substantial experimentation and technical expertise. Finite Element Analysis (FEA) software, such as ANSYS, COMSOL, and Abaqus, is fundamental for assessing the structural integrity of landing gear components, allowing detailed analysis of stress distribution, deformation, and even electromagnetic effects [91,95]. Multi-Body Dynamic (MBD) software, including RECURDYN V8R5, Adams 2012, and SIMSCAPE-MATLAB 2017b, is employed to simulate the complex motion of landing gear mechanisms, such as retraction, extension, and shock absorber behavior [96,97]. These tools are particularly valuable for demonstrating phenomena associated with aircraft touchdown, taxiing, and shimmy [98,99,100]. Collectively, they enable the creation of high-fidelity plant models that accurately represent the dynamic behavior of landing gear systems, allowing engineers to evaluate performance under realistic conditions and explore extensive design spaces with unprecedented precision.

To validate the mathematical model and assess the effectiveness of the control algorithm, many researchers have developed prototype landing gear drop tests [68,81,85,87,101,102]. In a typical drop-test experiment, sensors such as load cells, pressure sensors, and position sensors are installed to measure impact forces, hydraulic pressures, and motion, as illustrated in Figure 7. Experimental studies have produced compelling quantitative evidence demonstrating the superior performance of magnetorheological (MR) landing gear systems compared to passive counterparts, particularly in terms of damping force response time and energy absorption efficiency. A discrepancy between the damping force predicted by simulations and that observed in real-world drop tests can be seen in Figure 8 [87]. Notably, the difference between the mathematical model and experimental results is less than 10% during the initial peak period. Energy absorption efficiency—commonly quantified by shock absorber efficiency—serves as a primary metric for evaluating landing gear performance. Drop tests conducted in accordance with Federal Aviation Regulations [88] have revealed significant improvements for MR landing gear prototypes. Overall cushioning performance has been measured to be approximately 17.9% higher than that of passive systems in experimental drop tests of light aircraft prototypes [85]. These findings confirm that MR dampers effectively absorb substantial energy during landing, thereby enhancing structural integrity and reducing stress on the airframe.

4. Mathematical Model of MR for Landing Gear

Before building a semi-active vibration attenuation and control system for successful implementation, a precise mathematical model of a magnetorheological (MR) damper must be developed. Modeling MR dampers is a challenging undertaking due to their extremely nonlinear, bi-viscous, and hysteretic behavior, as shown in Figure 5. To capture these intricate characteristics for forecasting the damping force in various circumstances, several mathematical models have been put out to depict the dynamic behavior of MR dampers. The mathematical model could be classified as a pseudo-static model [47,103,104], parameter models [105,106], and unparameterized models [107]. The characteristics and applications of models are detailed in

Table 2. The comparison between the mathematical models of the MR damper.

Type	Characteristics	Yield Stress	Hysteresis	Computational Cost	Suitability for the MR Landing Gear
Pseudo-static models	Post-yield behavior with field-dependent yield stress	Yes	No	Low	Suitable for preliminary analysis, simulation, and real-time control design.
Parametric models	Internal hysteretic variable with a nonlinear evolution law	No	Yes	High	High-fidelity dynamic modeling and control performance assessment
Unparameterized model	Black-box mapping of inputs to force output	Yes (Implicit)	Yes (Implicit)	Very High	Suitable for adaptive control; limited physical interpretability

.

4.1. Pseudo-Static Models

Pseudo-static models determine the damper force by evaluating the pressure differential across the piston, which is governed by the yield stress as a function of magnetic field intensity, as mentioned in Section 2. Magnetorheological (MR) dampers generally operate in either the direct shear mode of the fluid or the pressure-driven (valve) flow mode [95]. The damper force associated with each mode is derived from distinct sets of governing equations as detailed in Figure 9. In the pressure-driven flow mode, the reduction in pressure is attributed to two principal mechanisms: viscous drag and field-dependent flow stress [108], as follows:

$$∆P={\displaystyle \frac{6\mu L}{\pi R_1{h}_1^3}}Q+c{\displaystyle \frac{l_p}{h_1}}{\tau }_c$$

(1)

Based on Equation (1), the hydraulic pressure on the up and down chambers is calculated and then multiplied by the corresponding cross-sectional area to create the total damping force, which can be described as a combination of the hydraulic force F_h and the pneumatic force F_a [109]:

$$F_d=F_a+F_H$$

(2)

where

$$\left\{\begin{array}{c}{F}_a=A_a\left[P_0{\left({\displaystyle \frac{V_0}{V_0-A_a·s}}\right)}^{po}-P_{ATM}\right]\\ F_h=A_p\left[{\displaystyle \frac{6\mu L}{\pi R_1{h}_1^3}}Q+c{\displaystyle \frac{l_p}{h_1}}{\tau }_c\right]\end{array}\right.$$

(3)

In Equation (3), the yield stress is developed based on the Bingham model [102,110] in Equation (4) or the Herschel–Bulkley model [111,112] as given in Equation (5):

$${\tau }_c={\tau }_{y}sign\left(\dot{\gamma }\right)+\mu \dot{\gamma }$$

(4)

$${\tau }_c={\tau }_y+k_h sign\left(\dot{\gamma }\right){\left(\dot{\gamma }\right)}^{n_h}$$

(5)

These models can represent shear-thinning or shear-thickening behavior. The relationship between yield stress (${\tau }_c$) and magnetic field strength is a fitting curve from the Lord company, as can be seen in Figure 5.

Although pseudo-static models can effectively describe the force-displacement relationship, they have significant limitations when it comes to accurately capturing the force-velocity characteristic. Furthermore, the multiple computational steps, cumulative errors, and computational costs increase. Therefore, they are more suitable for the design of magnetic (MR) dampers.

4.2. Parameter Model

To model the nonlinear and hysteresis behavior of magnetic (MR) dampers in landing gear systems, parametric modeling techniques are commonly used. These models simulate the physical characteristics of the damper under various operating settings using several mechanical and mathematical factors. The most common formulations for MR applications are the Bouc–Wen/Spencer hysteresis models, as can be seen in Figure 10. The complex relationship between force and displacement, as well as the velocity-dependent characteristics of MR fluids, is well captured by the Bouc–Wen model, particularly in its current-dependent form as given in Equation (6) [113]. Similarly, by incorporating both hysteresis and modulatory damping effects, the Spencer model provides a solid foundation for describing the dynamic response of MR dampers. Because they allow researchers to accurately predict system behavior under a wide range of landing conditions, these parametric models are crucial for accurate simulation, controller design, and performance optimization of smart landing systems [100,114]. The damping force of the MR damper can be calculated as follows:

$$\begin{gathered} F_d=c_0\left(\dot{x_1}-{\dot{y}}_1\right)+k_0\left(x_1-y_1\right)+k_1\left(x_1-x_0\right)+{\alpha }_h{z}_b \\ \dot{z_b}=-{\gamma }_h\left|\dot{x_1}-{\dot{y}}_1\right|z_b\mid z_b{\mid }^{n_h-1}-{\beta }_h\left(\dot{x_1}-{\dot{y}}_1\right)\mid z_b{\mid }^{n_h}+A_h\left(\dot{x_1}-{\dot{y}}_1\right) \end{gathered}$$

(6)

4.3. Unparameterized Model

Unparameterized modeling methods treat magnetic (MR) dampers as a “black box,” relying on data-driven techniques rather than physical assumptions about their internal structure. These models often employ mathematical approximations, such as neural networks, to describe the complex nonlinear relationship between input variables and the damping force [115,116,117,118]. A general representation of such models can be expressed as follows:

$$F_d=NN\left(x,\dot{x},\ddot{x},I\right)$$

(7)

The accuracy of nonparametric models depends heavily on the quality and quantity of training data. Data obtained from drop test prototypes often fail to cover all real-world landing scenarios, limiting the model’s generalizability. Furthermore, large datasets, while improving accuracy, significantly increase the complexity of the neural network architecture, leading to higher computational costs and longer processing times. These challenges have hampered the practical deployment of nonparametric models in real-time landing control systems, and as a result, their application remains largely limited to simulation environments rather than actual operational aircraft systems.

5. Aircraft MR Landing Gear Mathematical Model

Mathematical models of aircraft MR landing gear are crucial for analyzing and predicting their dynamic behavior during various ground operations, including landing, taxiing, and takeoff. These models help in optimizing design for safety, performance, and durability, and are essential for simulation, testing, and control system development. The complexity of these models varies depending on the specific phenomena being studied, such as shimmy, impact loads, vibration, or active control implementation [119].

5.1. One DOF

To investigate the dynamic behavior of an aircraft equipped with magnetorheological (MR) landing gear, a simplified one-degree-of-freedom (1-DOF) model is commonly adopted. Moreover, the MR dampers, which are equipped with a skid of a helicopter or a replacement shimmy damper in the nose landing gear, are normally modeled as a 1-DOF model. Figure 11 details the principle of the 1-DOF landing gear model. This model captures the essential vertical dynamics during landing impact while maintaining analytical tractability and suitability for control design. The mathematical model is detailed as follows:

$$F_d-M_{1}g=M_1{\ddot{z}}_1$$

(8)

The 1-DOF MR landing gear model provides valuable insight into landing impact attenuation, control law effectiveness, and parametric sensitivity [73,74,120]. However, it does not capture tire dynamics, structural flexibility, or multi-gear load redistribution. For comprehensive aircraft-level analysis, multi-degree-of-freedom models are required.

5.2. Two DOF

To better capture the dynamic interactions between the aircraft structure, landing gear strut, and tire–runway interface during touchdown and taxiing, a two-degree-of-freedom (2-DOF) sprung–unsprung mass model is widely employed, as shown in Figure 12. This model improves upon the 1-DOF representation by explicitly accounting for the unsprung mass of the landing gear and wheel assembly. Thus, the motion equations are given by:

$$\left\{\begin{array}{c}{F}_d-M_{1}g=M_1{\ddot{z}}_1\\ F_T-F_d=m_2{\ddot{z}}_2\end{array}\right.$$

(9)

Compared with the 1-DOF formulation, the 2-DOF model captures tire–runway interaction effects, allows assessment of wheel load and ground contact forces, and provides a more realistic framework for evaluating taxiing comfort and landing impact. The 2-DOF can be used to present the operation of the half main landing gear during the touchdown phase [75,85,87,121] and taxiing phase [122,123,124,125]. However, this model cannot provide the impact of pitch, roll, and yaw motion.

5.3. Full Aircraft Model

To accurately evaluate the performance of magnetorheological (MR) landing gear under realistic landing and taxiing conditions, a full aircraft dynamic model is required. Such a model accounts for the coupled vertical, pitch, and roll motions of the aircraft body and the interactions among multiple landing gear units, as can be seen in Figure 13. Depending on the number of landing gear and the type of aircraft, a full aircraft model has more than 6 DOF; in which the aircraft’s body have 3 linear motions follow 3 axis (x, y, z); and roll (φ), pitch (θ), and yaw (ψ) directions involves 6 DOF, each landing gear equipped with wheels have a 1 DOF (s₁, s₂, s₃). So, a small aircraft with 3 landing gears has a total of 9 DOF. However, it is hard to simulate all 9DOF under impact landing, which requires commercial software to simulate, such as RECURDYN [92,126], AMESim [127,128], and ADAMs [129,130,131]. Many other researchers reduce the DOF by abandoning yaw and x, y motion. So, the system involves only 6 DOF, as can be given in Equation (10):

$$\left\{\begin{array}{c}q={\left[\begin{array}{cccccc}{z}_0& s_1& s_2& s_3& \phi & \theta \end{array}\right]}^T\\ \ddot{q}=M_1^{-1}\left(Q_F-(f_L-M_1\ddot{q})\right)\end{array}\right.$$

(10)

where s₁, s₂, s₃ are the stroke of each landing gear; z0 is the displacement of aircraft’s body; φ is roll angle, θ is pitch angle.

This model does not account for the collision dynamics between the aircraft and the ground; rather, it is restricted to analyses of either the taxiing phase or the landing phase. During taxiing, the aircraft is assumed to move at a constant velocity while subjected to ground surface irregularities. In the landing phase, several simplified scenarios—such as single-point, two-point, and three-point landings—are typically modeled under the assumption of a flat runway surface [133,134,135]. However, the comprehensive landing process that incorporates the stochastic roughness characteristics of actual runways has not yet been systematically investigated.

6. Control Algorithms of the MR Landing Gear

Many control strategies are designed for the MR landing gear system, from classical control, such as bang-bang control and PID control, to modern control based on neural network control. All the control algorithms can be classified based on the taxiing phase and the touchdown phase. Because the control targets are different in different phases of the landing process, the researchers only focus on improving the riding comfort in the taxiing phase or improving landing performance in the touchdown phase. The controller strategies that were applied in full during the landing process have not been fully developed. The advantages and disadvantages of control algorithms are shown in

Table 3. Compare the control algorithm applied to the MR landing gear.

	Principle	Application	Advantages	Disadvantages
Bang Bang Control	Drive the damping force to a certain given value under various sink speeds	Touchdown phase [73,136]	- Simple to apply - Model-free control	- Limitation in efficiency - Requires an expensive load cell sensor - Lack of adaptiveness and robustness - High energy consumption
	Control input: $u(t)=\left\{\begin{array}{c}4A, F_{given}>F_{sen}\\ 0, F_{given}\le F_{sen}\end{array}\right.$
PID control	Drive the system to a reference acceleration.	Touchdown phase [137]	- Simple and easy to apply for both touchdown and taxiing	-Use a linear model, - Lack of adaptiveness and robustness
	Control input: $u(t)=\left\{\begin{array}{c}{K}_{p}e\left(t\right)+K_i\int e\left(t\right)+K_d\dot{e}\left(t\right), \dot{s}\dot{z_1}>0\\ \hfill 0, \dot{s}\dot{z_1}<0\end{array}\right.$
Skyhook control	Use dissipative devices to connect the aircraft body to an ideal fixed point in space	- Touchdown phase [75,83,84,138] - Taxiing phase [120,139]	- Simple and easy to apply for both touchdown and taxiing - Low energy consumption - Model-free control - Can be used in the taxiing phase	- Limitation in efficiency - Lack of adaptiveness and robustness
	Control input: $u(t)=\left\{\begin{array}{c}{C}_{sky}\dot{z_1}, \dot{s}\dot{z_1}>0\\ \hfill 0, \dot{s}\dot{z_1}<0\end{array}\right.$
Hybrid control	A combination of skyhook control and bang-bang control	- Touchdown phase [85,86,132,140]	- High efficiency under a certain condition - Low energy consumption	- Low adaptiveness and robustness - Required an accurate model
	Control input: $u(t)=F_{sky}+F_{desired}$ $F_{desired}=\left\{\begin{array}{c}{F}_{given}-F_{est}, F_{given}>F_{est}\\ \hfill 0, F_{given}\le F_{est}\end{array}\right.$
Sliding Mode Control	Drive the system according to the reference model	- Touchdown phase [48,96,141] - Taxiing phase [142,143]	- High robustness and adaptiveness - High shock absorber efficiency - Low energy consumption	- Required an accurate model - Complex to build an electrical board - High energy consumption
	Control input: $u\left(t\right)=\widehat{u}-k sat\left(S_f/\xi \right)$
H-infinity and LQR	Changes the system dynamics in order to obtain the gain required for the desired system response	- Touchdown phase [135,144,145] - Taxiing phase [124,133,146]	- Reduce the bounce of an aircraft to increase stability after landing - Can be applied in the taxiing phase	- Required an accurate model - Complex to build an electrical board
	$u=K_{LQR}x$
Mode Predictive Control	Use a model to predict the future behavior of the system	- Touchdown phase [147,148]	- High shock absorber efficiency - Low energy consumption - Can be used in the taxiing phase	- Required an accurate model - Complex to build an electrical board - Low robustness and adaptiveness
	Control input: $u(t)={\displaystyle \sum _{k=0}^N}Er(k)$ $Er(k)=\left\{\begin{array}{c}{F}_{given}-F_{est}, F_{given}>F_{est}\\ \hfill 0, F_{given}\le F_{est}\end{array}\right.$
Fuzzy Control	Used to adaptively adjust the base controller’s parameters, such as PID, bang-bang, skyhook, to directly generate the control output based on error and change in error signals	- Touchdown phase [121,149,150,151,152,153] - Taxiing phase [154]	- High adaptiveness - Low energy consumption - Average shock absorber efficiency - Model-free control - Can be used in the taxiing phase	- Complex to build an electrical board - Low robustness - Required a modest amount of experimental data
Neural network control	Train the neural network with experimental drop-test data under many conditions.	- Touchdown phase [93,101,155] - Taxiing phase [51,99,156,157]	- High adaptiveness - High shock absorber efficiency - Low energy consumption - Model-free control	- Required a huge experimental drop-test data - Low robustness - Cannot be used in the taxiing phase
	Control input: $u\left(t\right)=NN (x, \dot{x},\ddot{x})$

.

6.1. Taxiing Phase

There is an analogy between the aircraft’s landing gear and a car suspension, as can be seen in Figure 14. In both systems, the body (car or aircraft) moves independently of the tire that is in contact with the ground. Therefore, a two-degree-of-freedom (2-DOF) model, as presented in Equation (9), can be employed to describe the detailed dynamic behavior of the system. During the runway, the control’s goal is to reduce the vibration, but only for limited ground speeds and impact events [158,159]. The car suspension system is designed to absorb continuous road irregularities, provide comfort to passengers by smoothing bumps and vibrations during driving, and maintain tire-road contact for control and stability on rough terrain continuously [160,161,162]. Many researchers have attempted to use the control algorithm on a car’s suspension in the aircraft landing gear, which includes H-infinity control [124,135], backstepping controller [125], linear quadratic regulator (LQR) [32], skyhook control [122,123,139], fuzzy control [122,163], and an adaptive neuro-fuzzy inference system [155]. Similar to a car system [29,164,165], the root mean square (RMS) is also used to verify the effectiveness of the controller, as follows:

$$RMS\left(x\right)={\displaystyle \frac{1}{n}}\sqrt{\sum x^2}$$

(11)

where x is the signal of accelerometers or a position sensor. The reduction in RMS allows for the implementation of control algorithms similar to those employed in automotive suspension systems.

In some types of tricycle landing gear systems, a damper is equipped on the nose landing gear to prevent the shimmy phenomenon, which is the oscillating motion of the rudder assembly around the rudder axis, called the shimmy damper, as detailed in Figure 15 [166]. Shimmy dampers are designed to prevent or minimize side-to-side oscillations of the landing gear [98]. A small number of researchers developed the shimmy damper based on the MR damper to absorb vibration energy to inhibit shimmy. For example, Atabay and Ozkol simulated the free-play shimmy MR damper based on the Bonc–Wen model [113]; Dong et al. [33,167] developed the phase compensation active disturbance rejection control for the shimmy MR damper. Since the shimmy phenomenon is influenced by many factors related to the wheel, tire, and the landing gear structure itself, it is hard to build and verify a control strategy to completely eliminate the shimmy phenomenon.

Figure 14. The analogy between the aircraft’s landing gear and the car suspension (a) Car suspension MR damper modeling [168]. Reproduced with permission from Fernando Viadero-Monasterio, Miguel Meléndez-Useros, Manuel Jiménez-Salas, and Beatriz López Boada, Robust Static Output Feedback Control of a Semi-Active Vehicle Suspension Based on Magnetorheological Dampers, published by Applied Sciences, 2024; (b) Aircraft MR landing gear modeling [96].

Figure 14. The analogy between the aircraft’s landing gear and the car suspension (a) Car suspension MR damper modeling [168]. Reproduced with permission from Fernando Viadero-Monasterio, Miguel Meléndez-Useros, Manuel Jiménez-Salas, and Beatriz López Boada, Robust Static Output Feedback Control of a Semi-Active Vehicle Suspension Based on Magnetorheological Dampers, published by Applied Sciences, 2024; (b) Aircraft MR landing gear modeling [96].

Figure 15. Shimmy phenomenon and damper [169].

Figure 15. Shimmy phenomenon and damper [169].

6.2. Touchdown Phase

The landing gear system differs from the car suspension in that it deals with touchdown phases. The controller’s main goal is to absorb the landing impact during the crucial touchdown phase. That landing energy could adhere to the body’s structure and result in several dangerous circumstances. The FAR standard [4,88] is used to define the shock absorber efficiency in order to verify the landing performance of the landing gear:

$$\eta ={\displaystyle \frac{\int F_{d}ds}{\max F_d\max s}}$$

(12)

In Equation (12), the numerator of the equation represents the shock absorber’s ability to absorb energy, which can vary depending on the aircraft’s kinetic energy upon landing, while the denominator represents the maximum amount of energy the shock absorber can absorb. This efficiency is only defined after the aircraft has finished the touchdown. Therefore, there are many variations, such as reducing the maximum force acting on the fuselage, reducing the magnitude of acceleration, or reducing the maximum stroke. All these parameters cannot be measured in real time; they must be estimated or calculated in a very short time during the first stroke. It is the challenge of the developer to design control strategies that not only improve the impact performance but also adapt and remain stable under uncertainties and disturbances.

Bang-Bang control, which offers simplicity and model-free operation but suffers from inefficiency, high energy consumption, and poor adaptability, seeks to drive the damping force to a set value under varied sink speeds. Although it lacks robustness and adaptability, skyhook control uses dissipative devices to connect the aircraft body to an ideal fixed point, offering minimal energy consumption and ease of deployment. Skyhook and modified bang-bang techniques are combined in hybrid control to increase efficiency in some situations, but it still lacks resilience and requires precise modeling. Although it requires intricate electronics and accurate modeling, sliding mode control provides excellent robustness and efficiency by driving the system based on a reference model. Fuzzy control adaptively tunes base controllers using error signals, offering high adaptability and model-free operation, though it requires experimental data and has limited robustness. Lastly, neural network control leverages trained networks from drop-test data to generate control outputs, excelling in adaptability and efficiency but needing extensive data and suffering from low robustness. There are a few control strategies that can be applied to both taxiing and touchdown phases without modifying them too much, including skyhook and mode-predicted control. Other controls, such as hybrid control and sliding mode control, must be built into two different algorithms, then a switch mode is used to change after the first stroke. Additionally, intelligent control needs a lot of data in the taxiing phase to achieve good efficiency in this phase.

7. Challenges and Future Directions

Despite advancements in structural design and control strategies for aircraft landing gear systems using magnetorheological (MR) dampers, several critical challenges remain in methodology, implementation, and optimization.

Structural Design Limitations: Landing gear design has evolved from passive systems in small aircraft to configurations suitable for large fighter jets. However, durability assessment and long-term reliability testing remain underdeveloped. Applications in UAVs and airships are still largely unexplored.

Control Strategy Challenges: Existing strategies mainly enhance landing performance under varying sink rates. Yet, uncertainties such as temperature fluctuations and wind disturbances are often overlooked. MR fluid response delays during the touchdown phase—lasting only about 100 milliseconds—make real-time predictive control in multi-DOF systems extremely challenging. Furthermore, most approaches optimize either taxiing or touchdown, with few integrated frameworks covering the entire landing sequence.

Predictive Maintenance and AI Integration: Crash test and runway movement data have enabled machine learning models to adapt landing gear control under diverse conditions. Advanced predictive health management (PHM) techniques, leveraging AI and sensor networks, can monitor component health, predict failures, and enable proactive maintenance, improving safety and reliability.

Thermal and Rate-Dependent Effects: Temperature variations significantly alter MR fluid properties, requiring models that incorporate temperature-dependent parameters. During high-speed landings, rate-dependent phenomena and fluid compressibility become critical, demanding high-fidelity simulations for accurate dynamic response prediction.

8. Conclusions

This study has examined the state-of-the-art application of magnetorheological (MR) dampers in aircraft landing gear, with particular emphasis on structural design, mathematical modeling, and control strategies. The comparative analysis of MR landing gear configurations highlighted the diversity of design approaches and their implications for performance optimization. The investigation into mathematical models demonstrated that pseudo-static formulations are effective for damper design but insufficient for capturing dynamic force–velocity characteristics. Parameterized and unparameterized models, along with system representations ranging from single-degree-of-freedom (1-DOF) to full landing gear models, provide enhanced fidelity for control-oriented research. These modeling frameworks underscore the importance of balancing computational efficiency with accuracy in representing nonlinear and rate-dependent behaviors of MR fluids. Control strategies were classified and evaluated, revealing the potential of semi-active approaches to improve landing gear adaptability under varying operational conditions. However, limitations remain, particularly in the modeling of stochastic runway roughness and temperature-dependent fluid properties, which are critical for realistic aeronautical applications.

Overall, this research contributes to the consolidation of MR damper knowledge in aeronautical contexts by bridging structural design, modeling, and control perspectives. Future work should focus on integrating high-fidelity thermal and stochastic runway models, as well as exploring advanced control algorithms capable of real-time adaptation. Such developments will be essential for advancing MR damper technology toward more resilient, efficient, and adaptive landing gear systems in next-generation aircraft.